In our analysis we will consider the collinear PDFs
given by the CT14nnloIC parametrization [1]} from
a global analysis assuming that the \(x\) -dependence of
the intrinsic charm component is described by the BHPS model
[2]}. In this model the proton light cone wave function has
higher Fock states, one of them being \(|q q q c \overline{c}>\) .
The cross sections will be initially estimated in the next section using
the set obtained for \(P_{ic} = 1\%\)  and, for comparison, the results
for the case without IC will also be presented. Another important
ingredient is the modelling of \({\cal {F}}_{g^*} (x_2, k_t^2, \mu ^2)\) ,
which depends on the treatment of the QCD dynamics
for the unintegrated gluon distribution at small-\(x\) .
Currently, there are several models in the literature, some of them have
been reviewed in Ref. [3]}. In our analysis
we shall consider three different models:
two based on the solutions of linear evolution equations, which
disregard nonlinear (saturation effects) and one being the solution of
the Balitsky-Kovchegov equation [4]}, which takes into account
these effects in the small-\(x\)  regime.
In particular, we will use the uPDF derived using the Kimber-Martin-Ryskin (KMR) prescription [5]}, which assumes that the transverse
momentum of the partons along the evolution ladder is strongly ordered
up to the final evolution step. In the last step this
assumption breaks down and the incoming parton that enters into
the hard interaction posses a large transverse momentum (\(k_t \approx \mu \) ). Such prescription allow us to express \({\cal {F}}_{g^*} (x_2,k_t^2, \mu ^2)\)  in terms of Sudakov form factor, which resums all the
virtual contributions from the scale \(k_t\)  to the scale \(\mu \) , and a
collinear \(g\)  PDF, which satisfies the DGLAP evolution equations.
For this model, we will estimate the uPDF using as input the CT14nnlo
parametrization (with and without IC) [1]} and the associated
predictions will be denoted as KMR hereafter.
Some time ago we showed that in the case of charm production at the LHC,
the KMR uPDF leads to a reasonable description of the experimental data for
\(D\) -meson and \(D\bar{D}\) -pair production [7]}. As also
discussed in Refs. [8]}, [9]}, the KMR model
effectively includes extra emission of hard partons (gluons) from the
uPDF that corresponds to higher-order contributions and leads therefore
to results well consistent with collinear NLO approach.
In order to investigate the impact of new dynamical effects – beyond
those included in the DGLAP equation – that are expected to be present
in the small-\(x\)  regime, we will also estimate the charm cross section
using as input the uPDF's obtained in Ref. [10]} as
a solution of the Balitsky-Kovchegov equation [4]} modified to
include the sub-leading corrections in \(\ln (1/x)\)  which are given by a
kinematical constraint, DGLAP \(P_{gg}\)  splitting function and the
running of the strong coupling (for a detailed derivation see
Ref. [12]}). Such an approach includes the corrections
associated with the BFKL equation, in an unified way with the DGLAP one,
as well the nonlinear term, which takes into account unitarity
corrections.
In Ref. [10]} the authors performed a fit to the
combined HERA data and provided the solutions with and without the
inclusion of the nonlinear term. In the next section, we will use
these solutions as input in our calculations and the corresponding
predictions will be denoted KS nonlinear and KS linear, respectively.
For a comparison between predictions for the KMR, KS linear and KS
nonlinear \({\cal {F}}_{g^*} (x_2, k_t^2, \mu ^2)\)  we refer the interested reader
to Fig. 7 in Ref. [3]}.
